I am regressing firm characteristics on some stock trading-related measures in a panel dataset. Firm size is a highly significant control variable, independent of the estimation method etc. My focus variables are related to firm size though, either by construction (e.g. $focus variable = x / firmsize$) or because of an economic relationship.
As a consequence, I am finding myself in a classic multicollinearity situation: If firm size is put in as a control variable, my focus variables become insignificant. If firm size is left out, the focus variables are highly significant.
Any of the usual advice (e.g. http://en.wikipedia.org/wiki/Multicollinearity) is not helping: I cannot obtain more data, I cannot run my regression on principal components because I need interpretable coefficients etc.
I have little experience with this kind of problem but with some imagination, I came up with the following two ideas:
Running the regression with firm size as a control variables and additionally including interaction terms between each focus variable and firm size.
Trying to strip away the firm size effect from both the dependent variable and the focus variables, e.g. by first regressing firm size on the dependent/focus variable and then using the residuals as the dependent/focus variable in the actual regression.
Would either or both idea make any sense? Any comment or alternative ideas would be very welcome!